Annotation of src/lib/libm/src/e_pow.c, Revision 1.15
1.1 jtc 1: /* @(#)e_pow.c 5.1 93/09/24 */
2: /*
3: * ====================================================
4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5: *
6: * Developed at SunPro, a Sun Microsystems, Inc. business.
7: * Permission to use, copy, modify, and distribute this
1.11 simonb 8: * software is freely granted, provided that this notice
1.1 jtc 9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.10 lukem 13: #include <sys/cdefs.h>
1.7 jtc 14: #if defined(LIBM_SCCS) && !defined(lint)
1.15 ! lukem 15: __RCSID("$NetBSD: e_pow.c,v 1.14 2008/04/25 22:21:53 christos Exp $");
1.3 jtc 16: #endif
1.1 jtc 17:
18: /* __ieee754_pow(x,y) return x**y
19: *
20: * n
21: * Method: Let x = 2 * (1+f)
22: * 1. Compute and return log2(x) in two pieces:
23: * log2(x) = w1 + w2,
24: * where w1 has 53-24 = 29 bit trailing zeros.
1.13 drochner 25: * 2. Perform y*log2(x) = n+y' by simulating multi-precision
1.1 jtc 26: * arithmetic, where |y'|<=0.5.
27: * 3. Return x**y = 2**n*exp(y'*log2)
28: *
29: * Special cases:
30: * 1. (anything) ** 0 is 1
31: * 2. (anything) ** 1 is itself
32: * 3. (anything) ** NAN is NAN
33: * 4. NAN ** (anything except 0) is NAN
34: * 5. +-(|x| > 1) ** +INF is +INF
35: * 6. +-(|x| > 1) ** -INF is +0
36: * 7. +-(|x| < 1) ** +INF is +0
37: * 8. +-(|x| < 1) ** -INF is +INF
38: * 9. +-1 ** +-INF is NAN
39: * 10. +0 ** (+anything except 0, NAN) is +0
40: * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
41: * 12. +0 ** (-anything except 0, NAN) is +INF
42: * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
43: * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
44: * 15. +INF ** (+anything except 0,NAN) is +INF
45: * 16. +INF ** (-anything except 0,NAN) is +0
46: * 17. -INF ** (anything) = -0 ** (-anything)
47: * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
48: * 19. (-anything except 0 and inf) ** (non-integer) is NAN
49: *
50: * Accuracy:
51: * pow(x,y) returns x**y nearly rounded. In particular
52: * pow(integer,integer)
1.11 simonb 53: * always returns the correct integer provided it is
1.1 jtc 54: * representable.
55: *
56: * Constants :
1.11 simonb 57: * The hexadecimal values are the intended ones for the following
58: * constants. The decimal values may be used, provided that the
59: * compiler will convert from decimal to binary accurately enough
1.1 jtc 60: * to produce the hexadecimal values shown.
61: */
62:
1.4 jtc 63: #include "math.h"
64: #include "math_private.h"
1.1 jtc 65:
1.11 simonb 66: static const double
1.1 jtc 67: bp[] = {1.0, 1.5,},
68: dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
69: dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
70: zero = 0.0,
71: one = 1.0,
72: two = 2.0,
73: two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
74: huge = 1.0e300,
75: tiny = 1.0e-300,
76: /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
77: L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
78: L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
79: L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
80: L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
81: L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
82: L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
83: P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
84: P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
85: P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
86: P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
87: P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
88: lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
89: lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
90: lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
91: ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
92: cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
93: cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
94: cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
95: ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
96: ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
97: ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
98:
1.12 wiz 99: double
100: __ieee754_pow(double x, double y)
1.1 jtc 101: {
102: double z,ax,z_h,z_l,p_h,p_l;
1.14 christos 103: double yy1,t1,t2,r,s,t,u,v,w;
1.5 jtc 104: int32_t i,j,k,yisint,n;
105: int32_t hx,hy,ix,iy;
106: u_int32_t lx,ly;
1.1 jtc 107:
1.4 jtc 108: EXTRACT_WORDS(hx,lx,x);
109: EXTRACT_WORDS(hy,ly,y);
1.1 jtc 110: ix = hx&0x7fffffff; iy = hy&0x7fffffff;
111:
112: /* y==zero: x**0 = 1 */
1.11 simonb 113: if((iy|ly)==0) return one;
1.1 jtc 114:
115: /* +-NaN return x+y */
116: if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
1.11 simonb 117: iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
118: return x+y;
1.1 jtc 119:
120: /* determine if y is an odd int when x < 0
121: * yisint = 0 ... y is not an integer
122: * yisint = 1 ... y is an odd int
123: * yisint = 2 ... y is an even int
124: */
125: yisint = 0;
1.11 simonb 126: if(hx<0) {
1.1 jtc 127: if(iy>=0x43400000) yisint = 2; /* even integer y */
128: else if(iy>=0x3ff00000) {
129: k = (iy>>20)-0x3ff; /* exponent */
130: if(k>20) {
131: j = ly>>(52-k);
1.15 ! lukem 132: if((uint32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
1.1 jtc 133: } else if(ly==0) {
134: j = iy>>(20-k);
135: if((j<<(20-k))==iy) yisint = 2-(j&1);
136: }
1.11 simonb 137: }
138: }
1.1 jtc 139:
140: /* special value of y */
1.11 simonb 141: if(ly==0) {
1.1 jtc 142: if (iy==0x7ff00000) { /* y is +-inf */
143: if(((ix-0x3ff00000)|lx)==0)
144: return y - y; /* inf**+-1 is NaN */
145: else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
146: return (hy>=0)? y: zero;
147: else /* (|x|<1)**-,+inf = inf,0 */
148: return (hy<0)?-y: zero;
1.11 simonb 149: }
1.1 jtc 150: if(iy==0x3ff00000) { /* y is +-1 */
151: if(hy<0) return one/x; else return x;
152: }
153: if(hy==0x40000000) return x*x; /* y is 2 */
154: if(hy==0x3fe00000) { /* y is 0.5 */
155: if(hx>=0) /* x >= +0 */
1.11 simonb 156: return __ieee754_sqrt(x);
1.1 jtc 157: }
158: }
159:
160: ax = fabs(x);
161: /* special value of x */
162: if(lx==0) {
163: if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
164: z = ax; /*x is +-0,+-inf,+-1*/
165: if(hy<0) z = one/z; /* z = (1/|x|) */
166: if(hx<0) {
167: if(((ix-0x3ff00000)|yisint)==0) {
168: z = (z-z)/(z-z); /* (-1)**non-int is NaN */
1.11 simonb 169: } else if(yisint==1)
1.1 jtc 170: z = -z; /* (x<0)**odd = -(|x|**odd) */
171: }
172: return z;
173: }
174: }
1.11 simonb 175:
1.13 drochner 176: n = (hx>>31)+1;
177:
1.1 jtc 178: /* (x<0)**(non-int) is NaN */
1.13 drochner 179: if((n|yisint)==0) return (x-x)/(x-x);
180:
181: s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
182: if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
1.1 jtc 183:
184: /* |y| is huge */
185: if(iy>0x41e00000) { /* if |y| > 2**31 */
186: if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
187: if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
188: if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
189: }
190: /* over/underflow if x is not close to one */
1.13 drochner 191: if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
192: if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
1.11 simonb 193: /* now |1-x| is tiny <= 2**-20, suffice to compute
1.1 jtc 194: log(x) by x-x^2/2+x^3/3-x^4/4 */
1.13 drochner 195: t = ax-one; /* t has 20 trailing zeros */
1.1 jtc 196: w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
197: u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
198: v = t*ivln2_l-w*ivln2;
199: t1 = u+v;
1.4 jtc 200: SET_LOW_WORD(t1,0);
1.1 jtc 201: t2 = v-(t1-u);
202: } else {
1.13 drochner 203: double ss,s2,s_h,s_l,t_h,t_l;
1.1 jtc 204: n = 0;
205: /* take care subnormal number */
206: if(ix<0x00100000)
1.4 jtc 207: {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
1.1 jtc 208: n += ((ix)>>20)-0x3ff;
209: j = ix&0x000fffff;
210: /* determine interval */
211: ix = j|0x3ff00000; /* normalize ix */
212: if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
213: else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
214: else {k=0;n+=1;ix -= 0x00100000;}
1.4 jtc 215: SET_HIGH_WORD(ax,ix);
1.1 jtc 216:
1.13 drochner 217: /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
1.1 jtc 218: u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
219: v = one/(ax+bp[k]);
1.13 drochner 220: ss = u*v;
221: s_h = ss;
1.4 jtc 222: SET_LOW_WORD(s_h,0);
1.1 jtc 223: /* t_h=ax+bp[k] High */
224: t_h = zero;
1.4 jtc 225: SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
1.1 jtc 226: t_l = ax - (t_h-bp[k]);
227: s_l = v*((u-s_h*t_h)-s_h*t_l);
228: /* compute log(ax) */
1.13 drochner 229: s2 = ss*ss;
1.1 jtc 230: r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
1.13 drochner 231: r += s_l*(s_h+ss);
1.1 jtc 232: s2 = s_h*s_h;
233: t_h = 3.0+s2+r;
1.4 jtc 234: SET_LOW_WORD(t_h,0);
1.1 jtc 235: t_l = r-((t_h-3.0)-s2);
1.13 drochner 236: /* u+v = ss*(1+...) */
1.1 jtc 237: u = s_h*t_h;
1.13 drochner 238: v = s_l*t_h+t_l*ss;
239: /* 2/(3log2)*(ss+...) */
1.1 jtc 240: p_h = u+v;
1.4 jtc 241: SET_LOW_WORD(p_h,0);
1.1 jtc 242: p_l = v-(p_h-u);
243: z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
244: z_l = cp_l*p_h+p_l*cp+dp_l[k];
1.13 drochner 245: /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
1.1 jtc 246: t = (double)n;
247: t1 = (((z_h+z_l)+dp_h[k])+t);
1.4 jtc 248: SET_LOW_WORD(t1,0);
1.1 jtc 249: t2 = z_l-(((t1-t)-dp_h[k])-z_h);
250: }
251:
1.14 christos 252: /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
253: yy1 = y;
254: SET_LOW_WORD(yy1,0);
255: p_l = (y-yy1)*t1+y*t2;
256: p_h = yy1*t1;
1.1 jtc 257: z = p_l+p_h;
1.4 jtc 258: EXTRACT_WORDS(j,i,z);
1.1 jtc 259: if (j>=0x40900000) { /* z >= 1024 */
260: if(((j-0x40900000)|i)!=0) /* if z > 1024 */
261: return s*huge*huge; /* overflow */
262: else {
263: if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
264: }
265: } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
266: if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
267: return s*tiny*tiny; /* underflow */
268: else {
269: if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
270: }
271: }
272: /*
273: * compute 2**(p_h+p_l)
274: */
275: i = j&0x7fffffff;
276: k = (i>>20)-0x3ff;
277: n = 0;
278: if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
279: n = j+(0x00100000>>(k+1));
280: k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
281: t = zero;
1.4 jtc 282: SET_HIGH_WORD(t,n&~(0x000fffff>>k));
1.1 jtc 283: n = ((n&0x000fffff)|0x00100000)>>(20-k);
284: if(j<0) n = -n;
285: p_h -= t;
1.11 simonb 286: }
1.1 jtc 287: t = p_l+p_h;
1.4 jtc 288: SET_LOW_WORD(t,0);
1.1 jtc 289: u = t*lg2_h;
290: v = (p_l-(t-p_h))*lg2+t*lg2_l;
291: z = u+v;
292: w = v-(z-u);
293: t = z*z;
294: t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
295: r = (z*t1)/(t1-two)-(w+z*w);
296: z = one-(r-z);
1.4 jtc 297: GET_HIGH_WORD(j,z);
1.1 jtc 298: j += (n<<20);
299: if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
1.4 jtc 300: else SET_HIGH_WORD(z,j);
1.1 jtc 301: return s*z;
302: }
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